A rigid body spinning in space conserves kinetic energy T
and moment of momentum H.Expressions for the two are T=½(A w12 + Bw22
+ Cw32)
= const1, and H=Aw1i + Bw2j
+ Cw3k= const2 (a
vector constant in magnitude and direction).The reference frame i, j, k is body-fixed, so as
the body rotates the direction ofH is not constant in this frame.However the magnitude of H
must be constant, giving H=Ö(A2 w12 + B2w22
+ C2w32) = const2.Plotting constancy of T and H using w1w2 andw3
as ordinates gives two ellipsoids.The
only allowable spinning states are at their intersections.

Here is an example in which the momental ellipsoid is shown green/yellow/orange
and the energy ellipsoid is blue/white.The black line represents the intersection of the two ellipsoids, ie
the allowable states, showing that spin near the ‘B’ axis is a ‘saddle’ and is
unstable.You can check this by
spinning a book, a tennis racquet, a cellphone … anything really, and noticing
that it will not spin stably about the axis of its intermediate moment of
inertia.Spin about the other two axes
is stable.

To get a more complete picture, the momental ellipsoid shown
below is marked with the intersection lines of the energy ellipsoid for all
possible spins, the blue line being for spin about the ‘A’ axis (or close
thereto), the white being for spin about the ‘C’ axis – and note that both of
these are stable.The saddle for spin
about ‘B’ is the red line.(note:the colour coding of the ellipsoid is not
important – it is only intended to help give a 3D effect)

But spin about the ‘A’ axis is not really stable
because of energy dissipation effects.For a given moment of momentum H,the highest energy state is spin about the ‘A’ axis (the blue
line).Let us suppose that we spin the
body about axis ‘A’.It is possible to
envisage that as energy is lost within a spinning body (due to internal
friction, say) then the trajectory will gradually ‘spiral’ around the surface
of the ellipsoid, all the while conserving angular momentum H but losing energy
T.The spin will appear rather ‘wobbly’
as the body gradually settles down to spin about the lowest-energy state, about
the ‘C’ axis.There is no more energy
to lose now.

You can try this too.Take an object that will dissipate energy internally, for instance a key
wallet or a bag of frozen peas.Spin
this about any axis and it will quickly make its way to spinning about the axis
with largest moment of inertia – ie the minimum-energy state.